Model-theoretic Characterizations of Existential Rule Languages

Existential rules, a.k.a. dependencies in databases, and Datalog+/- in knowledge representation and reasoning recently, are a family of important logical languages widely used in computer science and artificial intelligence. Towards a deep understanding of these languages in model theory, we establish model-theoretic characterizations for a number of existential rule languages such as (disjunctive) embedded dependencies, tuple-generating dependencies (TGDs), (frontier-)guarded TGDs and linear TGDs. All these characterizations hold for arbitrary structures, and most of them also work on the class of finite structures. As a natural application of these characterizations, complexity bounds for the rewritability of above languages are also identified.Comment: 17 pages, 2 figures, the full version of a paper submitted to IJCAI 202

Distributed Graph Automata and Verification of Distributed Algorithms

Combining ideas from distributed algorithms and alternating automata, we introduce a new class of finite graph automata that recognize precisely the languages of finite graphs definable in monadic second-order logic. By restricting transitions to be nondeterministic or deterministic, we also obtain two strictly weaker variants of our automata for which the emptiness problem is decidable. As an application, we suggest how suitable graph automata might be useful in formal verification of distributed algorithms, using Floyd-Hoare logic.Comment: 26 pages, 6 figures, includes a condensed version of the author's Master's thesis arXiv:1404.6503. (This version of the article (v2) is identical to the previous one (v1), except for minor changes in phrasing.

Externalism, internalism and logical truth

The aim of this paper is to show what sorts of logics are required by externalist and internalist accounts of the meanings of natural kind nouns. These logics give us a new perspective from which to evaluate the respective positions in the externalist--internalist debate about the meanings of such nouns. The two main claims of the paper are the following: first, that adequate logics for internalism and externalism about natural kind nouns are second-order logics; second, that an internalist second-order logic is a free logic—a second order logic free of existential commitments for natural kind nouns, while an externalist second-order logic is not free of existential commitments for natural kind nouns—it is existentially committed

From IF to BI: a tale of dependence and separation

We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Vaananen, and their compositional semantics due to Hodges. We show how Hodges' semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O'Hearn, which combines intuitionistic and multiplicative connectives. This introduces several new connectives not previously considered in logics of informational dependence, but which we show play a very natural role, most notably intuitionistic implication. As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that they are adjoints to substitution, and hence uniquely determined. As for the dependence predicate, we show that this is definable from a simpler predicate, of constancy or dependence on nothing. This makes essential use of the intuitionistic implication. The Armstrong axioms for functional dependence are then recovered as a standard set of axioms for intuitionistic implication. We also prove a full abstraction result in the style of Hodges, in which the intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio

A Declarative Semantics for CLP with Qualification and Proximity

Uncertainty in Logic Programming has been investigated during the last decades, dealing with various extensions of the classical LP paradigm and different applications. Existing proposals rely on different approaches, such as clause annotations based on uncertain truth values, qualification values as a generalization of uncertain truth values, and unification based on proximity relations. On the other hand, the CLP scheme has established itself as a powerful extension of LP that supports efficient computation over specialized domains while keeping a clean declarative semantics. In this paper we propose a new scheme SQCLP designed as an extension of CLP that supports qualification values and proximity relations. We show that several previous proposals can be viewed as particular cases of the new scheme, obtained by partial instantiation. We present a declarative semantics for SQCLP that is based on observables, providing fixpoint and proof-theoretical characterizations of least program models as well as an implementation-independent notion of goal solutions.Comment: 17 pages, 26th Int'l. Conference on Logic Programming (ICLP'10

Foundational issues in implementing constraint logic programming systems

AbstractImplementations of Constraint Logic Programming (CLP) systems are often incomplete with respect to the theories they are intended to implement. This paper studies two issues that arise in dealing with these incomplete implementations. First, the notion of “satisfiability function” (the analogue of unification) is formally defined, and the question of which such functions are reasonable is studied. Second, techniques are given, based on the notion of satisfiability function, for formally (proof-theoretically) specifying an intended CLP theory or characterizing an existing CLP system. Such proof-theoretic characterizations have applications in proving soundness and completeness results, and proving properties of programs. Notions from substructural logic and the notion of Henkinness of the theory are shown to be important here

Derivation-Graph-Based Characterizations of Decidable Existential Rule Sets

This paper establishes alternative characterizations of very expressive classes of existential rule sets with decidable query entailment. We consider the notable class of greedy bounded-treewidth sets (gbts) and a new, generalized variant, called weakly gbts (wgbts). Revisiting and building on the notion of derivation graphs, we define (weakly) cycle-free derivation graph sets ((w)cdgs) and employ elaborate proof-theoretic arguments to obtain that gbts and cdgs coincide, as do wgbts and wcdgs. These novel characterizations advance our analytic proof-theoretic understanding of existential rules and will likely be instrumental in practice.Comment: accepted to JELIA 202

A Logical Characterization of Constant-Depth Circuits over the Reals

In this paper we give an Immerman's Theorem for real-valued computation. We define circuits operating over real numbers and show that families of such circuits of polynomial size and constant depth decide exactly those sets of vectors of reals that can be defined in first-order logic on R-structures in the sense of Cucker and Meer. Our characterization holds both non-uniformily as well as for many natural uniformity conditions.Comment: 24 pages, submitted to WoLLIC 202